Confidence intervals based on the intermediate approximation of theorem REF  generally fare poorly. We conjecture that this is so for two reasons. First, it is well-established in the EV theory literature that approximations based on intermediate order statistics suffer from bias when used to estimate quantiles (e.g. [1]}, [2]} and ch. 3 and 4 in [3]}). Including a correction term in the numerator might potentially improve coverages. Second, the rate of convergence in distribution for the distribution of the IVT is controlled by \(k=o(N)\) , and in samples considered here the values of \(k\)  available might be insufficient for normality to be a good approximation.
